Modelbased Manipulations of B Spline Surfaces to simulate Deformations of the Facial Tissue
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Modelbased Manipulations of B−Spline
Surfaces to simulate Deformations
of the Facial Tissue
D. Bartz
Computer Graphics Group and Telecommunications Institute
University of Erlangen, Germany
Outline:
• Overview
• Medical Foundations
• Modelbased Simulation
• Triangular B−Splines
• Results & Discussion
• Conclusions
Graphische Datenverarbeitung, Universität Erlangen−NürnbergOverview
Modelbased Simulation:
• mass spring system within five layers [Waters, Terzopoulos, 1990]
• manipulations of positions of nodes cause distortion
of equilibrium of springsystem
• calculate new equilibrium
• render top layer with spline method to improve resolution
• Waters, Terzopoulos: Manipulation of springs
with artifical muscle actors
• in this work: direct manipulation of spring nodes
within the layer, which represent bone layer
Graphische Datenverarbeitung, Universität Erlangen−NürnbergMedical Foundations (1)
Mechanical properties of facial tissue:
• epidermis (A)
• dermis (B)
• subcutis (C)
• fascia
• bone
Facial tissue has an inhomogeneous,
and non−directional behavior.
[From: PZ90]
Graphische Datenverarbeitung, Universität Erlangen−NürnbergMedical Foundations (2)
Human skull:
We manipulate the position
of the
• Maxilla
• Mandibular
[From: Rohe90]
Graphische Datenverarbeitung, Universität Erlangen−NürnbergModelbased Simulation (1)
• Each edge represents
a spring in a mass
spring system.
epidermis
• Spring stiffness dermis
with respect subcutis
to mechanical fascia
behavior
bone
3D laser scanner data: triangulation
Graphische Datenverarbeitung, Universität Erlangen−NürnbergModelbased Simulation (2)
Spring system after median cut
Graphische Datenverarbeitung, Universität Erlangen−NürnbergModelbased Simulation (3)
fi = mi . ai(t) + γi . vi(t) + gi , i ε {0..N}
where gi = Σc ij (||rij|| − lij) . rij
j ε Νi || rij ||
• equation describes balanced spring system
• moving of spring nodes causes distortion
• to compute new balance, use iteration method
Graphische Datenverarbeitung, Universität Erlangen−NürnbergModelbased Simulation (4)
ai(t) = (fi − gi . vi(t) − gi) . Bi/mi
• using a two step numerical integration method (Euler):
K1 = H(yn , tn) = a(vn, xn)
vn+1 = vn + ∆t . K1
xn+1 = xn + ∆t . vn+1
Graphische Datenverarbeitung, Universität Erlangen−NürnbergModelbased Simulation (5)
Algorithm:
determine processing steps and the
dedicated regions in bone layer
foreach processing step do
select all points in region
do
apply translation vector
to points
while points have not reached
target positions
od
Graphische Datenverarbeitung, Universität Erlangen−NürnbergTriangular B−Splines (1)
• surface: triangulation / unstructured grid
Tensor product−B−Splines not appropriate
Triangular B−Splines (DMS−Splines)
(Dahmen, Micchelli, Seidel 1992)
• properties:
• piecewise polynomial • affine invariant
• locality • convex hull
• positivity • local control
• optimal smoothness
Graphische Datenverarbeitung, Universität Erlangen−NürnbergTriangular B−Splines (2)
Mathematical foundations:
• based on
− Simplex Splines (Dahmen, Micchelli 1982)
− B−Patches (Seidel 1991)
• Computation of selected Simplex Splines, with respects to
a triangulation and a set of knot clouds (parametrization)
• half−open convex hull t2,2
t2,1
• barycentric coordinates t2,0
(positive oriented triangles)
t0,1
t0,0
t1,0
t0,2 t1,2
t1,1
Graphische Datenverarbeitung, Universität Erlangen−NürnbergTriangular B−Splines (3)
Four major problems to be solved:
• Construction of a proper parametrization
• Construction of a proper control net
• Construction of evaluation points in |R2
• Evaluation of Triangular B−Spline base functions at
the evaluation points
Graphische Datenverarbeitung, Universität Erlangen−NürnbergTriangular B−Splines (4)
Problem 1: parametrization
• Take advantage of the "scan property" of the triangulation in |R3
Cutting edge
Graphische Datenverarbeitung, Universität Erlangen−NürnbergTriangular B−Splines (5)
Construction of the knot cloud
foreach ti,0 e T do
if ti,0 on_rim_of_T then
estimate maximal_region m
estimate neighborhood of ti,0
select knot within m with best
quality concerning neighborhood
else
estimate neighborhood of ti,0
estimate knot with best quality With respects to:
concerning neighborhood • affine independent
fi • guaranteed partition
od of one
• no intersection of
edges
• no degenerated
triangles
Graphische Datenverarbeitung, Universität Erlangen−NürnbergTriangular B−Splines (6)
Problem 2: control net ... is an approximation problem:
• Simple approximation by vertex identification
Pros: fast, simple, robust
Cons: bad quality
• Least−Square−Approximation (LES)
Pros: better quality
Cons: high space and time complexity (O(n3))
(really huge design matrix)
Problem 3: Evaluation points
• n(n+1)/2 evaluation points per patch
• n=5 15 evaluation points:
Graphische Datenverarbeitung, Universität Erlangen−NürnbergTriangular B−Splines (7)
Problem 4: evaluation of base function
Algorithm
(Pfeifle, Seidel 1994) is
• a modification of
deBoor algorithm
• limited to quadric
Triangular B−Splines
C1 smoothness
Graphische Datenverarbeitung, Universität Erlangen−NürnbergResults & Discussion (1)
data set: P/[s] n=3 n=5 n=7
#vertices/ #vertices #vertices #vertices
#triangles E/[s] #triangles E/[s] #triangles E/[s] #triangles
A: 2115 8277 18487
552/1012 31 22 4048 94 16192 344 36432
B: 4586 18263 41032
1157/2273 68 60 9092 392 36368 1966 81828
C: 5979 23861 107136
1502/2976 102 193 11904 780 47616 3423 53647
D: 34282 136567 10000001
8641/17001 584 2517 47616 25156 272016 1500001 623036
Computed on a Onyx RealityEngine2, only using one processor (R4400)
1: estimate
Graphische Datenverarbeitung, Universität Erlangen−NürnbergResults & Discussion (2)
Problems:
• simplified tissue model:
− homogeneous conditionalization within every layer
− bone layer is approximated with a copy of a triangulation
of the epidermis layer
− no detail models (lips, eyes, ears,...)
− no muscle model (but easy to implement)
• results are strongly dependent on selection of vertices and translations
planning and reproduction is difficult
• method is not surgical oriented (at this stage)
• numerical problem with "bad" triangulations of the face
Graphische Datenverarbeitung, Universität Erlangen−NürnbergConclusion & Future Work
Conclusion:
• tissue model out of spring systems
• epidermis layer is rendered with Triangular B−Splines
Future Work:
• improvement of the tissue model
• improvement of the construction of the control net
• error correction of "bad" numerical data
• high potential in parallelization
Graphische Datenverarbeitung, Universität Erlangen−NürnbergReferences
DMS92 W. Dahmen, C. A. Micchelli, H.−P. Seidel: Blossoming begets
B−Splinebases built better by B−patches", In: Mathematics of
Computations. I(1), p. 97−115, July 1992
PS94a R. Pfeifle, H.−P. Seidel: "Faster evaluation of quadric bivariate
DMS−spline surfaces", Proc. of Graphics Interface’94,
p. 182−189, 1994
PZ90 W. Pschyrembel, C.Zink: "Pschyrembel Klinisches Woerter−
buch", Walter de Gruyter Verlag, Berlin, 1990
Rohe90 J.W. Rohen: "Funktionelle Anatomie des Menschen"
Schattauer Verlag, Stuttgart, 1990
Seide91 H.−P. Seidel: "Polar−Forms and Triangular B−Spline Surfaces",
In: Blossoming: The New Polar−Form Approach to Spline
Curves and Surfaces, SIGGRAPH’91, Course Note #26, 1991
WT90 K. Waters, D. Terzopoulos: "A physical model of facial tissure
and muscle articulation", In: Proc. of the First Conference on
Visualization in Biomedical Computing, p. 77−82, May 1990
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